Stay ahead with the world's most comprehensive technology and business learning platform.

With Safari, you learn the way you learn best. Get unlimited access to videos, live online training,
learning paths, books, tutorials, and more.

Chapter 12. Number Theory

Why is it that we entertain the belief that for every
purpose odd numbers are the most effectual?

—Pliny the Elder (23-79), Natural History

Number theory is thousands of years old. For most of those millennia,
it’s been “pure” mathematics, meaning that it had little
practical application. That has changed in the last few decades; many
of the most important advances in cryptography resulted from number
theory. In this chapter, we concentrate on prime numbers and modular
exponentiation, both of which are invaluable for the next
chapter: cryptography.

Before we can talk about prime numbers, we’ll first explore the basics
of number theory: the greatest common divisor (GCD)
and least common multiple (LCM), and techniques
for performing modular
arithmetic. Then we’ll use these techniques to find large prime
numbers. (We’ll also demonstrate caching techniques to make the search
faster.) Finally, we’ll spend a few pages on diversions: three simple
Perl programs that illustrate some unsolved problems in mathematics.

This is the most theoretical chapter in the book, but don’t let
that scare you. The mathematics involved is quite simple.

Basic Number Theory

Most of this section is about
divisors and
remainders. A divisor is a number which evenly
divides another number, its multiple. (2 is a
divisor of 6, 6 is a multiple of 2.) A
remainder is the amount left over when you divide one number into
another. (When you divide 7 by 2, the remainder is 1.)

We will generally ...

With Safari, you learn the way you learn best. Get unlimited access to videos, live online training,
learning paths, books, interactive tutorials, and more.